Matrices

An array of numbers arranged in a rectangular fashion and divided between rows and columns is called a matrix in mathematics. They are usually represented by writing all the numbers contained in them within square braces. There are many types of matrices and many operations like matrix multiplication which serve as crucial topics for boards and other entrance exams.

This is one of the most vital chapters in your maths syllabus. Almost all branches of studies that derive elements from mathematics, especially computer science, use this same concept thoroughly. For example, the figure below is that of a matrix with ‘m’ horizontal rows and ‘n’ vertical columns. 

(Image will be uploaded soon)

Different Types of Matrices

1. Column Matrix – A matrix that has elements only in one column is called a column matrix. 

\[\begin{bmatrix} 1\\ 0\\ -5 \end{bmatrix}\]

Figure 2: Column Matrix

2. Row Matrix – A matrix that has elements only in one row is called a row matrix. 

\[\begin{bmatrix} 1 & 5 & 9 \end{bmatrix}\]

Figure 3: Row Matrix

3. Invertible Matrix – A matrix A of size b x b is called an invertible matrix only when another matrix B exists of the same size such that AB = BA = I, where I is the identity matrix (containing only 1s in the principal diagonal) of the same dimension. In such a scenario, B is termed as the inverse matrix of A and also represented as A-1.

(Image will be uploaded soon)

Figure 4: Invertible Matrix

4. Singular Matrices – A matrix that has no inverse (from the previous definition) is called a singular matrix. The determinant value of the singular matrix is always 0. For example, the below matrix is singular because its determinant = 0.

For example:

\[\begin{pmatrix} 3 &12 \\ 2 & 8 \end{pmatrix}\]

The determinant is = (3 x 8) - (12 x 2)

= 24 - 24

= 0

Figure 5: Singular Matrix

5. Symmetric and Skew Symmetric Matrix – A matrix is called symmetric matrix if xij = xji, for all i and j, where xij = Element at ith row and jth column. Alternatively, a matrix is also called a symmetric matrix when its transpose is equal to the original matrix, AT=A. For example, the below matrix is symmetric because of the above conditions.

\[\begin{bmatrix} 3 & -2 & 4\\ -2 & 6& 2\\ 4& 2 & 3 \end{bmatrix}\]

Figure 6: Symmetric Matrix

A skew-symmetric matrix is a matrix that satisfies the condition, AT= -A. 

Pop Quiz 1

1. A matrix is a _______ array of numbers. 

1. Rectangular (Answer)

2. Square

3. Circular

4. None of the above

2. What is the most unique property of skew-symmetric matrices?

1. AT= A

2. AT= -A (Answer)

3. AT + A = I

4. AT. A = 0

Matrix Multiplication with a Scalar Number

A matrix can be multiplied with scalar numbers. If A = [aij]mxn (a matrix of size mxn) and k is a scalar which is to be multiplied to A, then the resultant matrix is obtained when each of the elements of A is multiplied with k, such that kA = [kaij]mxn. For example, take a look at the figure below. 

k\[\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}\]2x2  = \[\begin{bmatrix} ka_{11} & ka_{12}\\ ka_{21} & ka_{22} \end{bmatrix}\]2x2

Matrix Multiplication between Two Matrices

If A = [aij]m x n and B = [bij]n x p are two matrices such that the number of columns of A = number of rows of B, then the product of A and B is Cm x p. Each element cij of C is calculated with the formula below.  

\[C_{ij}\] = \[\sum_{h=1}^{n}a_{ik}b_{kj}\]

Properties for Multiplying Matrices

1. Multiplying two matrices can only happen when the number of columns of the first matrix = number of rows of the second matrix and the dimension of the product, hence, becomes (no. of rows of first matrix x no. of columns of the second matrix). 

2. In matrix multiplication, the order must be maintained as said in point #1. Without this order, multiplication cannot take place. 

3. In matrix multiplication, the associative rule states that (AB)C = A(BC).

4. In matrix multiplication, the commutative rule states that AB ≠ BA. 

Exercise

Take the following example and compute BC and A.(BC).

A = \[\begin{bmatrix} 1 & 0\\ 2 & 3\\ 3 & 1 \end{bmatrix}\]  B = \[\begin{bmatrix} 1 &2 & 1& 0\\ 0 & 1 & 0 & 2 \end{bmatrix}\]   C = \[\begin{bmatrix} 1\\ 1\\ 0\\ 1 \end{bmatrix}\]

Matrices are generally used in Geometry, but they are majorly used when the specification and representation of geometric transformation need to be done. For example, in rotations, coordinate changes and other activities. Whenever a numerical analysis is done, matrices play a vital role in its transformation. Solving computational problems is what matrices play a key role in. Matrices have a huge dimension and without them, many things might not be possible in mathematics. Other than geometry, there are other fields as well where matrices are taken into consideration.

So, this was all about matrices and all other operations and important types of them, which will be needed for your exams. To know more about other topics of mathematics, visit the Vedantu website or download the app today. We host such easy-to-read tutorials and other important guides there.